of 1 90
The Mathematical Nature Of AI and Life
By
Ian Beardsley
Copyright © 2021 by Ian Beardsley
of 2 90
Contents
Abstract……………………………..3
Important……………….…………..4
The Computation……….…………..5
The Dynamic Function……………..5
Free Electron Model………………..13
Band Theory……………………….14
Silicon And Carbon……..…………15
Germanium and Carbon………….17
Fundamental AI Bio Eqns…………19
Using Fundamenta Eqns………….20
Conclusion…………………………22
Semiconductor Devices……………24
The Mathematical Connection……26
The Broad Picture………………….30
Divergence………………………….31
The Heart of the Matter…………..39
The Planets…………………………45
The Cause of Inertia……………….46
Five-Fold Symmetry……………….51
Miller-Urey…………………………65
Bone…………………………………71
The Masculine and Feminine……..88
of 3 90
Abstract
I am finding mathematical connections between artificial intelligence and biological life and presenting
them as mathematical constructs. Of course if the structures of such elements are mathematical they can
be thought of as tensors making them invariant under transformation and thus Natural Law. However I
then proceed to to show what this mathematical structure is in terms of the modeling of such materials
with quantum chemistry to explain it. The QM result seems to make sense intuitively but much work
remains to be done. The proposition is speculative but compelling.
of 4 90
Important
Above we see the artificial intelligence (AI) elements pulled out of the periodic table of the elements. As
you see we can make a 3 by 3 matrix of them and an AI periodic table. Silicon and germanium are in
group 14 meaning they have 4 valence electrons and want 4 for more to attain noble gas electron
configuration. If we dope Si with B from group 13 it gets three of the four electrons and thus has a
deficiency becoming positive type silicon and thus conducts. If we dope the Si with P from group 15 it
has an extra electron and thus conducts as well. If we join the two types of silicon we have a
semiconductor for making diodes and transistors from which we can make logic circuits for AI.
As you can see doping agents As and Ga are on either side of Ge, and doping agent P is to the right of Si
but doping agent B is not directly to the left, aluminum Al is. This becomes important. I call (As-Ga) the
differential across Ge, and (P-Al) the differential across Si and call Al a dummy in the differential because
boron B is actually used to make positive type silicon.
That the AI elements make a three by three matrix they can be organized with the letter E with subscripts
that tell what element it is and it properties, I have done this:
Thus E24 is in the second row and has 4 valence electrons making it silicon (Si), E14 is in the first row
and has 4 valence electrons making it carbon (C). I believe that the AI elements can be organized in a 3 by
3 matrix makes them pivotal to structure in the Universe because we live in three dimensional space so
the mechanics of the realm we experience are described by such a matrix, for example the cross product.
Hence this paper where I show AI and biological life are mathematical constructs and described in terms
of one another.
We see, if we include the two biological elements in the matrix (E14) and and (E15) which are carbon and
nitrogen respectively, there is every reason to proceed with this paper if the idea is to show not only are
the AI elements and biological elements mathematical constructs, they are described in terms of one
another. We see this because the first row is ( B, C, N) and these happen to be the only elements that are
not core AI elements in the matrix, except boron (B) which is out of place, and aluminum (Al) as we will
see if a dummy representative, makes for a mathematical construct, the harmonic mean. Which means we
have proved our case because the first row if we take the cross product between the second and third rows
are, its respective unit vectors for the components, meaning they describe them!
E
13
E
14
E
15
E
23
E
24
E
25
E
33
E
34
E
35
of 5 90
The Computation
And silicon (Si) is at the center of our AI periodic table of the elements. We see the biological elements C
and N being the unit vectors are multiplied by the AI elements, meaning they describe them! But we have
to ask; Why does the first row have boron in it which is not a core biological element, but is a core AI
element? The answer is that boron is the one AI element that is out of place, that is, aluminum is in its
place. But we see this has a dynamic function.
The Dynamic Function
The primary elements of artificial intelligence (AI) used to make diodes and transistors, silicon (Si) and
germanium (Ge) doped with boron (B) and phosphorus (P) or gallium (Ga) and arsenic (As) have an
asymmetry due to boron. Silicon and germanium are in group 14 like carbon (C) and as such have 4
valence electrons. Thus to have positive type silicon and germanium, they need doping agents from group
13 (three valence electrons) like boron and gallium, and to have negative type silicon and germanium they
need doping agents from group 15 like phosphorus and arsenic. But where gallium and arsenic are in the
same period as germanium, boron is in a different period than silicon (period 2) while phosphorus is not
(period 3). Thus aluminum (Al) is in boron’s place. This results in an interesting equation.
A = (Al, Si, P )
B = (G a, G e, As)
A ×
B =
B
C
N
Al Si P
G a Ge As
= (Si ⋅ As − P ⋅ G e)
B + (P ⋅ G a − Al ⋅ As)
C + (Al ⋅ G e − Si ⋅ G a)
N
A = 26.98
2
+ 28.09
2
+ 30.97
2
= 50g /m ol
B = 69.72
2
+ 72.64
2
+ 74.92
2
= 126g /m ol
A ⋅
B = A Bcosθ
cosθ =
6241
6300
= 0.99
θ = 8
∘
A ×
B = A Bsi nθ = (50)(126)sin8
∘
= 877.79
877.79 = 29.6g /m ol ≈ Si = 28.09g /m ol
Si(A s − G a) + G e(P − Al )
SiG e
=
2B
Ge + Si
of 6 90
The differential across germanium crossed with silicon plus the differential across silicon crossed with
germanium normalized by the product between silicon and germanium is equal to the boron divided by
the average between the germanium and the silicon. The equation has nearly 100% accuracy (note: using
an older value for Ge here, is now 72.64 but that makes the equation have a higher accuracy):
Due to an asymmetry in the periodic table of the elements due to boron we have the harmonic
mean between the semiconductor elements (by molar mass):
This is Stokes Theorem if we approximate the harmonic mean with the arithmetic mean:
We can make this into two integrals:
If in the equation (The accurate harmonic mean form):
We make the approximation
28.09(74.92 − 69.72) + 72.61(30.97 − 26.98)
(28.09)(72.61)
=
2(10.81)
(72.61 + 28.09)
0.213658912 = 0.21469712
0.213658912
0.21469712
= 0.995
Si
B
(As − G a) +
Ge
B
(P − Al ) =
2SiG e
Si + G e
∫
S
( ∇ × u ) ⋅ d S =
∮
C
u ⋅ d r
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d xd y ≈
1
Ge − Si
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z ≈
1
3
1
(Ge − Si )
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Ge
B
(P − Al ) d x dz ≈
2
3
1
(Ge − Si )
∫
Ge
Si
yd y
Si
B
(As − G a) +
Ge
B
(P − Al ) =
Ge − Si
Ge
Si
dx
x
2SiGe
Si + Ge
≈ Ge − Si
of 7 90
Then the Stokes form of the equation becomes
Thus we see for this approximation there are two integrals as well:
For which the respective paths are
One of the double integrals on the left is evaluated in moles per grams, the other grams per mole
(0 to 1 moles per gram and 0 to 1 grams per mole).
The Geometric Interpretation…
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d yd z =
∫
Ge
Si
d x
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z =
1
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
Ge
B
(P − Al ) d ydz =
2
3
∫
Ge
Si
dz
y
1
=
1
3
B
SiGa
ln(z)
y
2
=
2
3
B
Si Al
ln(z)
of 8 90
of 9 90
By making the approximation
In
We have
is the differential across Si, is the differential across
Ge is the vertical differential."
Which is Ampere’s Circuit Law
We see if written
Which is interesting because it is semiconductor elements by molar mass, which are used to
make circuits.
We say (Phi) is given by
and
And
=1.618
=0.618
(phi) the golden ratio conjugate. We also find
2SiGe
Si + Ge
≈ Ge − Si
Si(As − G a)
B
+
Ge(P − Al )
B
=
2SiGe
Si + Ge
Si
ΔGe
ΔS
+ G e
ΔSi
ΔS
= B
ΔSi = P − Al
ΔGe = As − Ga
ΔS = Ge − Si
∇ ×
B = μ
0
J + μ
0
ϵ
∂
E
∂t
Si
ΔGe
ΔS
= B − Ge
ΔSi
ΔS
Φ
a = b + c
a
b
=
b
c
Φ = a /b
ϕ = b /a
ϕ
(ϕ)ΔGe + (Φ)ΔSi = B
of 10 90
Thus since
And we have
We see and are both and c is in the Si (silicon) field wave, but for E and B fields c is the speed of
light.
To find the Si wave our differentials are
∇ ×
B = μ
J + μϵ
0
∂
E
∂t
Si
ΔGe
ΔS
= B − Ge
ΔSi
ΔS
ΔGe =
ΔS
Si
B −
Ge
Si
ΔSi
(
∇
2
−
1
c
2
∂
2
∂t
)
E = 0
(
∇
2
−
1
c
2
∂
2
∂t
)
B = 0
c =
1
ϵ
0
μ
≈ ϕ
μ
ϵ
0
Φ
ϕ
ϵ
0
= 8.854E − 12F ⋅ m
−1
μ = 1.256E − 6H /m
Ge
Si
= μϵ
0
ΔS
Si
= μ
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Si = 0
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Ge = 0
of 11 90
It is amazing how accurately we can fit these differentials with and exponential equation for the upward
increase. The equation is
This is the halfwave:
ΔC = N − B = 14.01 − 10.81 = 3.2
Δ Si = P − Al = 30.97 − 26.98 = 3.99
ΔG e = A s − G a = 74.92 − 69.72 = 5.2
Δ Sn = Bi − In = 121.75 − 114.82 = 6.93
ΔPb = Bi − T l = 208.98 − 204.38 = 4.6
y(x) = e
0.4x
+ 1.7
y(x) = e
2
5
x
+
17
10
y(x) = e
0.4x
+ 1.7
y(x) = e
2
5
x
+
17
10
of 12 90
Interestingly, the 0.4 is boron (B) over aluminum (Al) the very two elements that lead us to looking for a
wave equation because boron was the out of place element in the AI periodic table that lead to us using
aluminum as its dummy representative in the Si differential and that itself divided into the left hand terms
to give us the harmonic mean between the central AI elements semiconductor materials Si and Ge. The
Ag and Cu are the central malleable, ductile, and conductive metals used in making electrical wires to
carry a current in AI circuitry.
y(x) = e
B
Al
x
+
Ag
Cu
B
Al
=
10.81
26.98
= 0.400667
Ag
Cu
=
107.87
63.55
= 1.6974 ≈ 1.7
of 13 90
Free Electron Model
Metals are held together by free electrons that move throughout the solid. The free electron
model views these electrons as a gas. For the one-dimensional case a line through which the
electrons move and have no collisions with one another which is a one-dimensional square well
whose walls are the edge of a rod and respects the exclusion principle; no two electrons in an
orbit can have the same state, there can only be spin up and spin down. For the special case
where its temperature T is 0 degrees K the the energy levels are filled with electrons from the
from lowest to highest two at a time until the highest energy level is filled. This highest energy
level filled is the Fermi energy . Moving to the three dimensional case with electrons moving
freely in a metal block we consider it a three-dimensional infinite square well and solve the
Schrodinger equation:"
"
Where the wave functions of the electron states are:"
"
Three L’s with sides x,y,z, and three n’s x,y,z, for respective quantum numbers corresponding
to motions in the x,y,z directions. The allowed energies are:"
"
The number of of conduction electrons per unit volume which is the number of filled states
per unit volume is the same as the number of electrons per unit volume. The Fermi energy is:"
"
We are interested in Aluminum which is 18.1E28 electrons per cubic meter gives a Fermi
energy in electron-volts of 11.7 eV. Aluminum is part of our equation (the dummy in the Si
differential) that produces Stokes theorem:"
"
Or,…"
E
F
−
h
2
2m
e
(
∂
2
∂x
+
∂
2
∂y
+
∂
2
∂z
)
ψ (x, y, z) = E ψ (x, y, z)
ψ (x, y, z) =
2
L
x
sin
n
x
π x
L
x
2
L
y
sin
n
y
π x
L
y
2
L
z
sin
n
z
π x
L
z
π
2
h
2
2m L
2
(
n
2
1
+ n
2
2
+ n
2
3
)
n
e
E
F
=
h
2
2m
e
(
3π
2
N
V
)
2
3
∫
1
0
∫
1
0
[
Si
B
(As − Ga) +
Ge
B
(P − Al)
]
dydz =
∫
Ge
Si
d x
of 14 90
"
"
Where and "
Band Theory
A conductor is different than an insulator in how its electrons respond to an electric
field. In a conductor a large amount of electrons respond whereas electrons in
insulators belong to completely filled bands. We are interested in silicon and
germanium as well and we have only considered aluminum a conductor.
Semiconductors are like insulators except in that they have a small energy gap
between the filled band and the next unavailable unfilled band. The bands at T=0 K are
completely filled in both insulators and semiconductors, however at room temperature
the energy gap is small enough that that electrons can escape from the valence band
to the conduction band and they can respond to an applied field. This band gap in
silicon is 1.14 eV and in germanium is 0.67 eV. In GaAs is 1.43 eV, and in GaP is 2.26
eV at 300K."
Thus, we have if we take gallium arsenide band gap in eV to be the differential As-Ga
across Ge in molar mass…"
From"
"
"
And, this is…"
"
"
"
∫
1
0
∫
1
0
Si
B
(As − Ga)dydz =
1
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
Ge
B
(P − Al)dydz =
2
3
∫
Ge
Si
dz
1/3 ≈ 1 − ϕ
2/3 ≈ ϕ
∫
1
0
∫
1
0
Si
B
(As − Ga)dydz =
1
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
Si
B
(Ga As)dydz =
∫
1
0
∫
1
0
1.14eV
B
(1.43eV )dydz
1
3
∫
Ge
Si
dz =
1
3
(0.67 − 1.14)
(−0.1567eV )B = 1.6302eV
2
B = (10.4eV )
of 15 90
The Boltzmann constant is "
This times room temp in kelvin 300K yields 0.02585eV is the kinetic energy of a particle
at room temperature to get an idea of the order of energies we are discussing. The
Fermi energy for Au=5.53 eV and for Ag=5.49 eV. If we divide the Fermi energy of
aluminum by that of gold or silver we see they are about half that of aluminum. The
interesting thing here is that B=10.40333/2 is 5.20eV is close to
where 8.298eV is the first ionization energy of boron and "
B/2=5.20eV is approximately the activation energy of boron which is 4.7eV the energy
required for a reaction. If we take 2/3 of 8.298eV we have 5.532eV. This averaged with
4.7 eV is 5.116eV is almost exactly our B/2=5.20eV. It makes sense we divide by two
because this is the average between boron band gap and ionization energy which is
1.5eV plus 8.298eV equals 5.6eV is close to 5.20eV. In this sense we see AI as a
mathematical construct in terms of molar mass is in correlation with the quantum
mechanical model of these elements in terms of Fermi energies, band gap energies
and dissociation energies, as well as ionization energies and activation energies. While
the amount of doping agents used to dope semiconductors is extraordinarily small, the
effect is enormous, increasing the conductivity significantly. We have said biological life
describes AI elements in the three by three AI matrix that is conveniently pulled out of
the periodic table, and that this has a dynamic function utilizing boron and aluminum.
But just what are the connections of these mathematical constructs to biological life
elements. We address that here,…"
Silicon and Carbon
We guess that artificial intelligence (AI) has the golden ratio, or its conjugate in its means
geometric, harmonic, and arithmetic by molar mass by taking these means between doping
agents phosphorus (P) and boron (B) divided by semiconductor material silicon (Si) :
Which can be written
k
B
= 8.617E − 5eV ⋅ K
−1
ϕ(8.298eV ) = 5.128eV
PB
Si
=
(30.97)(10.81)
28.09
= 0.65
2 PB
P + B
1
Si
=
2(30.97)(10.81)
30.97 + 10.81
1
28.09
= 0.57
0.65 + 0.57
2
= 0.61 ≈ ϕ
of 16 90
We see that the biological elements, H, N, C, O compared to the AI elements P, B, Si is the
golden ratio conjugate (phi) as well:
So we can now establish the connection between artificial intelligence and biological life:
Which can be written:
Where HNCO is isocyanic acid, the most basic organic compound. We write in the arithmetic
mean:
Which is nice because we can write in the second first generation semiconductor as well
(germanium) and the doping agents gallium (Ga) and arsenic (As):
Where
Where ZnSe is zinc selenide, an intrinsic semiconductor used in AI, meaning it doesn’t require
doping agents. We now have:
PB(P + B) + 2PB
2(P + B)Si
≈ ϕ
C + N + O + H
P + B + Si
≈ ϕ
(P + B + Si )
PB(P + B) + 2PB
2(P + B)Si
≈ (C + N + O + H )
PB
[
P
Si
+
B
Si
+ 1
]
+
2 PB
P + B
[
P
Si
+
B
Si
+ 1
]
≈ 2HCNO
[
PB +
2 PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ 3HNCO
[
PB +
2 PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ HNCO
[
Ga
Ge
+
As
Ge
+ 1
]
Zn
Se
≈
[
P
Si
+
B
Si
+ 1
]
[
Ga
Ge
+
As
Ge
+ 1
]
PB
(
Zn
Se
)
+
2 PB
P + B
(
Zn
Se
)
+
P + B
2
(
Zn
Se
)
≈ HNCO
of 17 90
Germanium And Carbon
We could begin with semiconductor germanium (Ge) and doping agents gallium (Ga) and
Phosphorus (P) and we get a similar equation:
,
In grams per mole. Then we compare these molar masses to the molar masses of the
semiconductor material Ge:
Then, take the arithmetic mean between these:
We then notice this is about the golden ratio conjugate, , which is the inverse of the golden
ratio, . . Thus, we have
1.
2.
This is considering the elements of artificial intelligence (AI) Ga, P, Ge, Si. Since we want to find
the connection of artificial intelligence to biological life, we compare these to the biological
elements most abundant by mass carbon (C), hydrogen (H), nitrogen (N), oxygen (O),
phosphorus (P), sulfur (S). We write these CHNOPS (C+H+N+O+P+S) and find:
A similar thing can be done with germanium, Ge, and gallium, Ga, and arsenic, As, this time
using CHNOPS the most abundant biological elements by mass:
2GaP
Ga + P
= 42.866
GaP = 46.46749
2GaP
Ga + P
1
Ge
=
42.866
72.61
= 0.59
GaP
1
Ge
=
46.46749
72.61
= 0.64
0.59 + 0.64
2
= 0.615
ϕ
Φ
ϕ ≈
1
Φ
GaP(Ga + P) + 2GaP
2(Ga + P)Ge
≈ ϕ
GaP(Ga + P) + 2GaP
2(Ga + P)Si
≈ Φ
CHNOP S
Ga + As + Ge
≈
1
2
[
Ga As +
2Ga As
Ga + As
+
Ga + As
2
][
Ga
Ge
+
As
Ge
+ 1
]
≈ CHNOPS
[
Ga
Si
+
As
Si
+ 1
]
of 18 90
We can also make a construct for silicon doped with gallium and phosphorus:
And for germanium doped with gallium and phosphorus:
Here is a table of the AI biological equations…
Ga As
(
O
S
)
+
2Ga As
Ga + As
(
O
S
)
+
Ga + As
2
(
O
S
)
≈ CHNOPS
O
S
≈
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga
Si
+
As
Si
+ 1
]
Ga As(G a + As) + 2G a As
2(Ga + As)Ge
≈ 1
C + H + N + O + P + S
Ga + As + Ge
≈
1
2
(C + N + O + H ) ≈
2(Ga + P)Si
GaP(Ga + P) + 2GaP
(P + B + Si )
HNCO ≈
2(Ga + P)Si
(Ga + P)
[
GaP +
2GaP
Ga + P
]
(P + B + Si )
HNCO ≈
2(P + B + Si )Si
GaP +
2GaP
Ga + P
GaP(Ga + P) + 2GaP
2(Ga + P)Ge
≈ ϕ
[
GaP +
2GaP
Ga + P
+
Ga + P
2
][
P
Ge
+
B
Ge
+
Si
Ge
]
≈ HNCO
[
Ga
Ge
+
As
Ge
+ 1
]
GaP
(
B
S
)
+
2GaP
Ga + P
(
B
S
)
+
Ga + P
2
(
B
S
)
≈ HNCO
of 19 90
The Fundamental AI Bioequations
[
PB +
2 PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ HNCO
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga As +
2Ga As
Ga + As
+
Ga + As
2
][
Ga
Ge
+
As
Ge
+ 1
]
≈ CHNOPS
[
Ga
Si
+
As
Si
+ 1
]
[
GaP +
2GaP
Ga + P
+
Ga + P
2
][
P
Ge
+
B
Ge
+
Si
Ge
]
≈ HNCO
[
Ga
Ge
+
As
Ge
+ 1
]
HNCO ≈
2(P + B + Si )Si
GaP +
2GaP
Ga + P
PB(P + B) + 2PB
2(P + B)Si
≈ ϕ
Ga As(G a + As) + 2G a As
2(Ga + As)Ge
≈ 1
GaP(Ga + P) + 2GaP
2(Ga + P)Ge
≈ ϕ
GaP(Ga + P) + 2GaP
2(Ga + P)Si
≈ Φ
C + N + O + H
P + B + Si
≈ ϕ
C + H + N + O + P + S
Ga + As + Ge
≈
1
2
Zn
Se
≈
[
P
Si
+
B
Si
+ 1
]
[
Ga
Ge
+
As
Ge
+ 1
]
O
S
≈
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga
Si
+
As
Si
+ 1
]
of 20 90
Using The Fundamental Equations
Now that we have outlined the fundamental AI Bioequations, let us put them to use. We
consider:
Making the approximations: ,
, we obtain:
Which can further be written by saying :
Which is interesting because the Si times itself is then equal to something times itself in that Ge
and Si are both semiconducting materials, but Ge is larger than Si, however this is compensated
for by reducing it by a factor of the golden ratio conjugate, phi. The equation is however only
79% accurate because there has been a lot of drift due to so many approximations. However if
we reduce phi by a factor of itself and write:
It is then 99% accurate:
If we do the same with the other and write:
(P + B + Si)
PB(P + B) + 2PB
2(P + B)Si
≈ (C + N + O + H )
HNCO ≈
2(P + B + Si)Si
GaP +
2GaP
Ga + P
GaP ≈ ϕGe
2GaP
Ga + P
≈ ϕG e
PB ≈ ϕSi
2Si
2
Ge
= ϕSi +
2P B
P + B
2P B
P + B
≈ ϕSi
Si
2
= ϕGeSi
Si
2
= ϕ
2
GeSi
28.09 = (72.64)(28.09)(0.381924) = 27.9g/m ol
27.916
28.09
= 0.99
2Si
2
Ge
= ϕ
2
Si +
2P B
P + B
of 21 90
We have:
Which is better but still only 81% accurate. However if we write it:
Then it is 95.87% accurate. But we see in the first approximation that . That is we
have boron, the element that is out of place in the AI periodic table resulting in the dynamics of
our equations. So, we can write…
This gives…
Which is 26.836 which is close to aluminum (Al=26.98) which is the dummy representative for
boron in our equations. We have incredibly:
With an accuracy of nearly 100%. This becomes…
While phosphorus, boron, silicon, and germanium and gallium and arsenic are the primary AI
elements, gold (Au), Silver (Ag) and copper (Cu), are the fundamental AI elements in that they
conductive, ductile, and malleable. Incredibly, the number 3 in the above equation is the ratio of
gold to copper in molar mass, so we have…
21.72 = (0.381924)(28.09) + 16.026 = 10.72 + 16.026 = 26.75
2Si
2
Ge
= ϕ
3
Si +
2P B
P + B
phi
2
Si ≈ B
2Si
2
Ge
= B +
2P B
P + B
10.81 + 16.02 = B +
2P B
P + B
Al = B +
2P B
P + B
Al = B
3P + B
P + B
Al = B
Au
Cu
P + B
P + B
Au
Cu
=
196.97
63.55
= 3.099 ≈ 3
of 22 90
Conclusion
Since we have"
"
"
"
And "
"
"
We have"
"
"
We also have that"
"
∫
1
0
∫
1
0
[
Si
B
(As − Ga) +
Ge
B
(P − Al)
]
dydz =
∫
Ge
Si
d x
∫
1
0
∫
1
0
Si
B
(As − Ga)dydz =
1
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
Ge
B
(P − Al)dydz =
2
3
∫
Ge
Si
dz
C + N + O + H
P + B + Si
≈ ϕ
PB(P + B) + 2PB
2(P + B)Si
≈ ϕ
∫
1
0
∫
1
0
Ge
B
(P − Al)dydz =
C + N + O + H
P + B + Si
∫
Ge
Si
dz
∫
1
0
∫
1
0
Ge
B
(P − Al)dydz =
PB(P + B) + 2PB
2(P + B)Si
∫
Ge
Si
dz
Si
ΔGe
ΔS
= B − Ge
ΔSi
ΔS
of 23 90
Which is a form of Maxwell’s equation:"
"
Where"
"
Making"
"
Become"
"
Because"
"
Where Ge/Si are not equal to mu times epsilon not, but play the role of that product.
The golden ratio conjugate plays the role of c the speed of light. This gives the
halfwave"
"
And we have said"
"
This is interesting because the Fermi energy for gold (Au) is 5.53eV and for silver (Ag) is
5.49 eV. This is on the order of our B/2=5.20eV. Gold and silver are the best electric
conductors and as such are used in AI prolifically. Copper is a very good conductor as
∇ ×
B = μ
J + μϵ
0
∂
E
∂t
Ge
Si
= μϵ
0
(
∇
2
−
1
c
2
∂
2
∂t
)
E = 0
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Si = 0
c =
1
ϵ
0
μ
≈ ϕ
y(x) = e
B
Al
x
+
Ag
Cu
Al = B
Au
Cu
P + B
P + B
of 24 90
well but is used the most because it is cheap and abundant. I believe this relevant in
light of "
"
"
The Fermi energy of Aluminum (Al) is 11.7eV that of copper (Cu) is 7.00 eV. The fermi
energies of Au and Ag added together are that of Al."
Au+Ag=Al=5.49+5.53=11.02eV~11.7eV"
Semiconductor Devices
Coulomb’s Law:"
"
Which states the force exerted between two charges q1 and q2 are directly
proportional to the product of their charges, and inversely proportional to the square of
the distance between them. "
The charge of a proton is the same as the charge of an electron, except for the proton
the charge is positive and for the electron it is negative. Like charges repel and
opposite charges attract. The charge of a proton and of an electron is .
The constant of proportionality in Coulomb’s Law is:"
"
The equation of force of a charge moving in an electric field and magnetic field is
given by the Lorenz Force:"
"
The Hall effect is the production of a potential due to the motion of a conductor in a
magnetic field. The Hall voltage ( ) created by the drift velocity v of an electron in a
magnetic field B through a conductor of width w is#
#
"
y(x) = e
B
Al
x
+
Ag
Cu
Al = B
Au
Cu
P + B
P + B
F = k
e
q
1
q
2
r
2
1.6 × 10
−19
C
k
e
=
1
4πϵ
0
= 9 × 10
9
E
B
F = q(
E +
v ×
B )
V
H
V
H
= vBw
of 25 90
An n-type semiconductor is created by doping a semiconductor material such as
silicon valence 4- with a doping agent or, donor, of an extra electron such as arsenic
valence 5-. Such a doped substance has its electrons easily excited from the valence
band to the conduction band by a small amount of heat. Easily because the extra
electron has no positive charges to keep it in. As these electrons move into the
conduction band, they leave behind holes which can be thought of as a current as well
because they have electrons falling into them. This current is said to be carried by
minority carriers, and the current due to electrons in the conduction band is said to be
that of majority carriers. In a p-type semiconductor the majority carries are said to be
the free holes because in this case the material is doped with something like gallium
valence -3 which one electron less than the silicon valence 4-. Minority carriers are
then holes left behind due to thermal excitation of electrons in the valence band across
the gap to the conduction band."
If we put a semiconductor strip in a magnetic field (field lines at right angles to its
surface) it drives positive holes to its upper end which gather there until the electric
field acting along the surface (right angles to the magnetic field) balances with the force
due to the magnetic field:"
"
"
And we have "
The drift velocity v, of the majority carriers can thus be found and is:"
"
Or we can get it from the Hall voltage. From this we can determine the current density j
as well:"
"
Where is the number of charge carriers per unit volume, q is the charge of an electron,
and v is the drift velocity. The flow of electrons into the conduction band equals the
holes left behind, or n=p"
Semiconductor devices like diodes and transistors are created when we connect n-
type semiconductors with p-type semiconductors. Thousands, even millions of these
device can be formed on a small piece of silicon with connections between the
conducting paths."
F
B
= qvB
F
E
= qE
E = vB
v =
E
B
j = nq v
of 26 90
For diodes n-type and p-type semiconductors are joined with negative donors on the p
side and positive acceptors on the negative side at the p-n junction. In the reverse bias
configuration the + side of a battery is applied to the n-type side of the diode which
widens the depletion layer increasing the potential energy across the p-n junction. As
such few electrons get through to establish a current. However if the + side of the
battery is connected to the p-type material the depletion layer is narrowed, the
potential across the junction decreases and electrons flow from one side to the other.
This makes the diode a one-way valve."
The fraction of electrons that have enough energy to diffuse across the potential barrier
is given by the Maxwell-Boltzmann distribution. In the reverse bias hookup it is:"
"
And much better in the forward bias connection with voltage hooked up in that bias:"
"
is the current with no voltage applied across the diode, just the minority carriers,
which is independent of the bias voltage. Just the flow of electrons from the valence
band to the conduction band due to heat. We have the net current is:"
"
To make a transistor we join three semiconductors so we can have npn or pnp
transistors. These are three lead diodes. I if we apply a voltage to the the middle p in
the npn transistor, to the base, we have a junction transistor which is not just a one
way valve like the diode, but that can be turned on or off to control the current flow.
Hooking up a positive terminal to the base causes a small current to flow which in turn
causes a large current to flow at the collector (n) from the emitter (the other n). By
controlling the base current we can control the collector current. The transistor is like a
valve and can be used to either amplify currents with a continuous analog signal or
switch currents either on or off so we can make switches that encode in binary making
computers possible, on or off, 1 or zero."
The Mathematical Connection
We can greatly simply the problem if we want to search for an explanation for AI as a
mathematical construct by looking at diodes instead of transistors. While logic gates
are built so we can have AI by forming millions of interconnected transistors on silicon
crystals and interconnecting them, we can build all of the logic gates with diodes; the
AND gate, OR gate, NOT gate…"
Ne
eV/k
B
T
V
B
I = I
0
e
eV
B
/k
B
T
I
0
I
net
= I
0
(
e
eV
B
/k
B
T
− 1
)
of 27 90
Indeed early on, in first developing AI we made the equivalent of diodes with vacuum
tubes. They had the same elements and principles behind them. In this case they were
tungsten filaments in a glass (SiO2) tube. Electrons emitted by thermionic emission of
heating the filament is like the emitter in a np diode, is the cathode. Electrons are
attracted by a metal plate, the anode like the collector in the np diode. In a triode we
have a metal screen or grid placed between the cathode and anode which can control
the current by voltage applied to it like the base in a npn transistor."
We are looking for some characteristics that characterize germanium and silicon
diodes, if we want explain our equations that characterize them as mathematical
constructs. We begin like this…"
In the reverse bias configuration for extreme values of reverse bias the material ionizes
causing an avalanche of current which occurs at breakdown voltage. The current is
this the reverse saturate current."
The Hall effect is best described as the electric field E that arises if a metal or
semiconductor carrying a current I is placed in a magnetic field of flux density B which
is developed along a direction perpendicular to B and I."
The values we are working with are:"
=diode current"
=diode reverse saturation current"
V=External voltage applied to diode"
=a constant 1 for germanium 2 for silicon"
is the Thermal Voltage"
=Boltzmann Constant is 1.380649E-23 J/K"
charge of electron (1.60217662E-19C)"
T=temperature of diode junction in degrees kelvin"
Room Temperature 300K is "
Our equation is:"
"
I
0
−I
sat
I
I
0
ξ
V
T
= K T/q = T/11600
k
B
q =
V
T
= 26mv
I = I
0
(
e
(V/ξV
T
)
− 1
)
of 28 90
…"
The one value that is constant for both Si and Ge diodes is the their forward biases, the
voltage that turns them on. For Si this is 0.7V for Ge this is 0.3V. We say"
"
"
T=6960 degrees Kelvin (12068.33 degrees F)"
T=3480 degrees Kelvin (5804.33 degrees F)"
At these temperatures the currents generated in Si and Ge diodes without being
connected to a voltage could turn on another set of Si and Ge diodes."
We can also write "
So we have"
"
"
Silicon: "
Germanium: "
Thus if V over V thermal is in the ratio of Silicon forward bias to germanium forward
bias we have factors of 1.718 and 6.389 for constants of Si=2 and Ge=1,
respectively. We see that gold, the heaviest fine conductor divided by germanium the
heaviest fine semiconductor is"
(196.97g/mol)/(72.64g/mol)=2.71~e=2.718"
And, the second heaviest fine conductor, silver, divided by silicon the second heaviest
fine semiconductor is"
(107.87g/mol)(28.09g/mol)=3.840~e+1=3.718"
0.6V = K T/q = T/11600
0.3V = K T/q = T/11600
V/V
T
= 0.6V/0.3V
I
I
0
= e
2/2
− 1
I
I
0
= e
2/1
− 1
I
I
0
= e − 1 = 1.718
I
I
0
= e
2
− 1 = 6.389
ξ
of 29 90
"
"
The lightest fine conductor is copper. As we progress from germanium, to silicon, the
next, and lighter element in that group is carbon, which though is not a fine
semiconductor, it is the core element to biological life. We have:"
(63.55g/mol)/(12.01g/mol)=5.2914"
"
It is at this point that we remember"
"
We bring this up because Boron out of place in the AI periodic table being with the
biological elements, aluminum in its place use as the dummy differential across Si,
could be substituted with gallium (Ga) even though it belongs to the differential across
Ge because it is symmetrical placed with boron (B) around aluminum (Al). We have"
"
This is in our halfwave"
"
That comes from"
"
Where"
I
I
0
=
Au
Ge
− 1
I
I
0
=
Ag
Si
− 2
I
I
0
=
Cu
C
− e = 2.57 ≈
Ga
Al
=
69.72
26.98
= 2.584
Si
B
(As − Ga) +
Ge
B
(P − Al) =
2SiGe
Si + Ge
B
Al
=
10.81
26.98
= 0.40
y(x) = e
B
Al
x
+
Ag
Cu
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Si = 0
of 30 90
"
The Broad Picture
We see as we progress from the parallel placements (Au/Ge) through the table to (Ag/
Si) and finally to (C/Cu) where we have run out of slots, we progress from AI to AI and
finally Life (C/Cu). But what does the fine electric conductor of AI, which is Cu, have to
do with carbon and biological life?"
The biochemistry of all terrestrial life seems to be similar throughout the spectrum of
species, and so much so that it would seem we all descended from a common
ancestor. Of course at the most fundamental level life as we know it is based on
carbon. This is because it has 4 valence electrons allowing it to combine in long chains
with hydrogen called hydrocarbons, and combine readily with oxygen 2-, and nitrogen
3-. Biologist have hazarded to look at silicon as a candidate since it has a valence of 4
as well, and came to the conclusion that life did not form like this because in the
presence of oxygen it readily combines with it making SiO2 silicon dioxide the basic
ingredient of sand. Perhaps many different organisms originated differently and
independently but at some point we may have a common ancestor. Her name is LUCA
(Last Universal Common Ancestor). Using a comparative approach biochemists have
traced our ancestry back to LUCA who shared the traits we have today. Some of these
traits are she stored her genetic information in DNA, and made use of the same twenty
amino acids we use to make our proteins with the same RNA machinery and genetic
code we use."
However, in dating LUCA we looked at the metabolic pathways common to all of life
and found her metabolism was based on iron (Fe). In today’s oxygen rich environment
iron quickly oxidizes to its ferric state and is highly insoluble. It is believed then that
LUCA lived before the earth was rich in oxygen. Today we use in our metabolic
pathways copper (Cu) because its oxidized state is more soluble than its reduced
forms. Marine organisms use iron and it is the limiting nutrient in the ocean and the
marine organisms have developed mechanisms with which to to extract iron form
bacteria. But the important point here is that the earth became oxygen rich with the
arrival of photosynthesizers, plants that could use energy from the sun to make
electrochemical energy (sugars) from carbon dioxide and in the process make oxygen.
The oxygen rich atmosphere seems to have happened about 2 billion years ago but the
evidence for life on earth goes back as far as 3.5 billion years, putting LUCA’s age in a
wide gap of 2 billion to 3.5 billion years old.$
ϕ = 0.618 =
1
Ge /Si
=
1
2.58597
= 0.62185
of 31 90
Divergence
We did a stokes theorem formulation of the AI elements. We want to do a divergence theorem
formulation. Our stokes theorem formulation was"
"
We want to consider"
"
We have to find the F-vector. It is from"
"
, , "
"
"
"
We have:"
"
We have said the speed of light in electrodynamics is given by"
"
∫
1
0
∫
1
0
Si
B
(As − Ga)dydz =
1
3
∫
Ge
Si
dz
∫∫∫
V
∇ ⋅
F dV =
∫∫
S
(
F .
n)d
S
i
j
k
∂
∂x
∂
∂y
∂
∂z
0
Si
B
Ga ⋅ z
Si
B
As ⋅ y
=
Si
B
(As − Ga)
i
F
x
= 0
F
y
=
Si
B
Ga ⋅ z
F
z
=
Si
B
As ⋅ y
∇ ⋅
F = 0
∫
V
0dV = constant
∫∫
S
(
F
x
, F
y
, F
z
)
⋅
(
dydz, d xdz, d xdy
)
=
Si
2B
(
Ga ⋅ z
2
+ As ⋅ y
2
)
Si
2B
(
Ga ⋅ z
2
+ As ⋅ y
2
)
=
Q
ϵ
0
c =
1
ϵ
0
μ
of 32 90
And in our formulation of the AI elements plays the role of the speed of light, is the
golden ratio conjugate is:"
"
Thus"
"
"
Thus we see"
"
"
Thus in our formulation the golden ratio conjugate plays the role of the speed of light in
electrodynamics. This makes sense because in relativity theory velocities are
considered a fraction of the speed of light because the speed of light is the fastest
speed attainable. The speed v squared over the speed of light squared is subtracted
from 1. That is the Lorentz contraction is:"
"
See the following storyboard pages to the computations…"
ϕ
ϕ =
1
Ge /Si
Ge /Si = μϵ
0
ΔS
Si
= μ =
Ge − Si
Si
= Φ
ϵ
0
=
Ge
Ge − Si
= Φ
1
μϵ
0
=
1
Φ
2
= ϕ
L = L
0
1 −
v
2
c
2
of 33 90
$
of 34 90
$
of 35 90
$
of 36 90
$
of 37 90
$
of 38 90
We see the eccentricity of the ellipse "
"
Plays the role of the beta factor in"
"
In the equation"
"
We see the eccentricity is"
"
Which is 9 figures eight of which are the first eight consecutive integers 1,2,3,4,5,6,7,8.
The only number that occurs twice is 2. Kind of interesting."
1 −
D
C
L = L
0
1 −
v
2
c
2
Si
2B
(
Ga ⋅ z
2
+ As ⋅ y
2
)
=
Q
ϵ
0
1 −
Ga
As
= 1 −
69.72
74.92
= 0.263452781
of 39 90
The Heart of the Matter
If we take our halfwave:"
"
And apply the schrodinger equation:"
"
Then,…"
"
"
We know that"
"
Or,…"
"
We notice"
"
And"
"
This says"
"
The molar masses of Ge and Si are such that the geometric mean between Ge and Si is the
difference between Ge and Si. This has solutions:"
$
y(x) = e
B
Al
+
Ag
Cu
(
∂
2
∂x
−
1
ϕ
2
∂
2
∂x
)
(
e
B
Al
x
)
= 0
(
B
Al
)
2
(
1 −
1
ϕ
2
)
= 0
ϕ = 1
ϕ =
1
Φ
ϕΦ = 1
Ge
Ge − Si
=
72.64
72.64 − 28.09
= 1.63 ≈ Φ
Si
Ge − Si
=
28.09
72.64 − 28.09
= 0.63 ≈ ϕ
GeSi = Ge − Si
y =
1
2
(3x − 5x)
y =
1
2
(3x + 5x)
of 40 90
If x=1 we have:"
0.381, 2.618"
If x=2 we have"
0.76, 5.236"
If x=3 we have"
1.145898, 7.854"
If x=4 we have"
1.52786, 10.47"
If x=5 we have"
1.9098, 13.090"
This is a pair of intersecting lines:"
"
"
With slopes"
"
"
The plots are…$
y = (1 − ϕ)x = 0.38x
y = (1 + Φ)x = 2.62x
m
1
= (1 − ϕ)
m
2
= (1 + Φ)
of 41 90
$
of 42 90
Since"
"
"
But we have said:"
"
But we also have"
"
This says"
"
"
Or…"
0=1"
Since we have said"
"
Is"
"
Then we might guess"
"
Is 1 is E and is c and is m in"
1 − ϕ =
Si
Ge
=
28.09
72.64
= 0.3867
1 + Φ =
Ge
Si
=
72.61
28.09
= 2.58597
Ge
Si
=
1
ϕ
1 − ϕ =
Si
Ge
Φϕ
2
= 1
1
ϕ
(ϕ − 1) = 1
1 −
Ga
As
1 −
v
2
c
2
Φϕ
2
= 1
ϕ
Φ
of 43 90
"
Which makes sense because"
"
Is the analog of"
"
In electrodynamics and, "
"
Is the analog of Maxwell’s equation"
"
And,…"
"
Is the analog of"
"
But that 0=1 may be that the quantum states 0 and quantum states 1 are
interchangeable in Hund’s rule and allow for more than 2 electrons in a bond. And we
see Alexander got to the heart of the matter when he wrote concerning my earlier work
in this project:"
Александр Сергеевич (Alexander) of Institute of Physical Chemistry (Moscow) wrote "
E = mc
2
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Si = 0
(
∇
2
−
1
c
2
∂
2
∂t
)
E = 0
ΔGe =
ΔS
Si
B −
Ge
Si
ΔSi
∇ ×
B = μ
J + μϵ
0
∂
E
∂t
c =
1
Ge
Si
c =
1
ϵ
0
μ
of 44 90
Your approach is very relevant and touches on one of the main provisions of quantum
chemistry-namely, the well-known Hund’s rule that sets the number of electrons
involved in the formation of a bond (usually n=2). However, it is still not clear why only
two electrons must necessarily participate in the formation of a chemical bond - there
are no prohibitions in quantum mechanics in this case! Perhaps your proposed
classification confirms the possibility of participation in the formation of a chemical
bond of several electrons, more than two in number! Then it is possible to substantiate
the presence of some criterion that makes it possible to distinguish between
biologically active and passive elements. And this is extremely important for
understanding how life arose on Earth!!
of 45 90
The Planets
If our equations the solutions of the wave describe any kind of quantization, it is that of
the first 7 planets:"
"
Substituting n for x, the planetary number we have…"
n=1 yields 0.381"
Mercury is 0.387AU"
n=2 yields 0.76"
Venus is 0.72AU"
n=3 yields 1.146"
Earth is 1.00AU"
n=4 yields 1.528"
Mars is 1.52Au"
n=5 yields 1.91"
Asteroids are 2-3AU"
Then on the other side of the asteroid belt we switch to the other equation that is a
solution:"
"
n=2 yields 5.236"
Jupiter is 5.2AU"
n=4 yields 10.47"
Saturn is 9.5AU$
y =
1
2
(3x − 5x)
y =
1
2
(3x + 5x)
of 46 90
The Cause of Inertia
I had been working on a theory of biological life and AI as interconnected in physical
structure and as mathematical constructs. At some point I had to think of a way to carry
this to the most basic level of matter as a mathematical construct. That is relationship
between point, plane and line. To do this I had to say it was nothing, though had
properties. To do this I thought of it as the three dimensional cross-section of a four
dimensional hypersphere. However in the end if we want to use G the universal
gravitational constant, and coulomb’s constant for electric charge, we are essentially
asking for a reconciliation between gravity and quantum mechanics. I find the only way
it can be done is through the fine structure constant. There is nothing new here in that it
is believed this mysterious number, the most accurately determined experimentally, if
were slightly different stars would not fuse hydrogen, helium and beryllium into carbon
and there would then not be life in the universe. It is fundamental to everything and has
lead to an anthropic principle suggesting the universe is fine-tuned, and that ultimately
it would be at the heart the long sought reconciliation between gravity theory and
quantum mechanics. Further the number is determined experimentally not derived
mathematically, and governs the interaction of charged particles with electromagnetic
radiation.
of 47 90
Abstract
Matter is that which has inertia. This means it resists change in position with a force
applied to it. The more of it, the more it resists a force. We understand this from
experience, but what is matter that it has inertia?
I would like to answer this by considering matter in one of its simplest manifestations,
the proton, a small sphere with a mass of 1.6726E-27 kg. This is a measure of its inertia.
I would like to suggest that matter, often a collection of these protons, is the three
dimensional cross-section of a four dimensional hypersphere.
The way to visualize this is to take space as a two-dimensional plane and the proton as a
two dimensional cross-section of a sphere, which would be a circle.
In this analogy we are suggesting a proton is a three dimensional bubble embedded in a
two dimensional plane. As such there has to be a normal vector holding the higher
dimensional sphere in a lower dimensional space. Thus if we apply a force to to the cross-
section of the sphere in the plane there should be a force countering it proportional to
the normal holding it in a lower dimensional universe. This counter force would be
experienced as inertia. It may even induce in it an electric field, and we can see how it
may do the same equal but opposite for the electron. Refer to the illustration on the
following page…
of 48 90
of 49 90
If and
This is about
F = G
Mm
r
2
G = 6.67408E − 11
N ⋅ m
2
kg
2
r
p
= 0.833E − 15m
r
e
= 2.817940E − 15m
m
p
= 1.67262E − 27kg
m
e
= 9.1E − 31kg
h = 6.62607E − 34J/s
c = 299792458m /s
F = m a
m =
F
a
F
n
= μFcosθ
m =
μh 4π r
2
cosθ
Gc
μ = 1
θ = 0
∘
m =
h 4π r
2
Gc
m =
(6.62607E − 34J/s)4π (0.833E − 15m)
2
(6.67408E − 11Nm
2
/kg
2
)(299792458m /s)
m = 5.373681E − 31kg
5.373681E − 31
1.67262E − 27
= 0..00032
of 50 90
Approximately three ten thousandths the mass of a proton. If we divide that by the fine
structure constant squared, we have
6.0 protons even, the number of protons in carbon, the central atom to biological life. If
and (Giving equal weight to x and y components) then
We have
protons
About 5 protons evenly which is Boron (B). If we try 60 degrees then
In order for this to be 4 protons evenly we must have
We have used here the fine structure constant squared, which is often used as well as to
the minus 1. The fine structure constant squared is the ratio of the potential energy of an
electron in the first circular orbit to the energy given by the mass of an electron in the
Bohr model times the speed of light squared:
and may open up a lot of doors.
α =
1
4πϵ
0
e
2
ℏc
=
1
137
α
2
=
1
18769
m = (0.00032)(18769) = 6.0
μ = 1
θ = 45
∘
m = 2.8876E − 61(0.7071) = 4.51865E − 31kg
4.51865E − 31
1.67262E − 27
= 0..00027
n = (0.00027)(18769) = 5
m = 2.8876E − 61(0.5) = 3.7997E − 31kg
3.7997E − 31
1.67262E − 27
= 0..000227
m = (0.000227)(18769) = 4.26
μ = 0.93896
α
2
=
U
e
m
e
c
2
μ
θ
of 51 90
Five-Fold Symmetry
To make artificial intelligence (AI) we need semiconductors, like diodes and transistors. To
make semi conductors we need to dope Silicon Si 4- with a group 13 doping agent to have
positive silicon such as with boron B 3- or with a group 15 doping agent like phosphorus P 5-
to have negative type silicon. Or we can dope germanium Ge 4- with a group 13 doping agent
like gallium Ga 3- for positive type germanium or with a group 15 doping agent like arsenic As
5- to have negative type silicon. We connect the negative with the positive to have a
semiconductor, meaning a current can run through it in only one direction. "
We pull these AI elements out of the periodic table of the elements to make an AI periodic
table:"
We now notice we can make a 3 by 3 matrix of it, which lends itself to to the curl of a vector
field, by including biological elements carbon C (above Si):"
="
="
="
"
Let us dot this with and take the double integral over Si to Ge
over both variable sets:"
i
j
k
∂
∂x
∂
∂y
∂
∂z
(C ⋅ P)y (Si ⋅ Ga)z (Ge ⋅ As)y
(Ge ⋅ As − Si ⋅ G a)
i + (C ⋅ P)
k
[
(72.64)(74.92) − (28.09)(69.72)
]
i +
[
(12.01)(30.97)
]
k
3,482
(
g
mol
)
2
i + 372
(
g
mol
)
2
k
(
zdyd z
i + yd x d y
k
)
d xdy
of 52 90
="
="
="
="
"
"
Now let us take the harmonic mean between Si and Ge. It is"
"
And the arithmetic mean between them:"
"
We see the value of 44.3 g/mol is somewhere between the harmonic and arithmetic mean.
Perhaps it is the geometric mean…"
"
Thus we can say…"
∫
Ge
Si
∫
Ge
Si
(
3,483
(
g
mol
)
2
i + 372
(
g
mol
)
2
k
)
⋅
(
zdyd z
i + yd x d y
k
)
∫
Ge
Si
∫
Ge
Si
(
3,483
(
g
mol
)
2
⋅ zdzd y + 372
(
g
mol
)
2
⋅ yd x d y
)
∫
Ge
Si
3,483
(
(72.64 − 28.09)
2
2
)
dy +
∫
Ge
Si
372y ⋅ (72.64 − 28.09)d y
3456359
(
g
mol
)
4
(72.64 − 28.09) + 16573
(
g
mol
)
3
(
(72.64 − 28.09)
2
2
)
170427030.8
(
g
mol
)
5
5
170427030.8 = 44.3g/mol
2SiGe
Si + Ge
= 40.5g/mol
Si + Ge
2
= 50.365g/m ol
SiGe = 45g/mol
of 53 90
"
"
Which like Stoke’s Theorem in that it relates an integral of a flux over a surface to path integral.
The expression on the right-hand side of the equation is the geometric mean between Si and
Ge. This integral can better be represented with product calculus:"
"
Where and and n=2. If we we say the arithmetic mean is A, and the harmonic
mean is H, the geometric mean G…"
"
This is"
"
This is quite interesting because"
"
"
"
I say interesting because we can write all three of these as one equation, the f-mean:"
"
The harmonic mean and the arithmetic mean are special cases of the power-mean which is the
case when , the harmonic mean when p=-1, and the arithmetic mean when p=1."
u = (CP ⋅ y, SiG a ⋅ z, Ga As ⋅ y)
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = exp
(
1
Ge − Si
∫
Ge
Si
ln(x)d x
)
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
x
1
= Si
x
2
= Ge
A + H
2
= 45.4325 ≈ G
Si
2
+ 6SiGe + Ge
2
4(Si + Ge)
≈ SiGe
H(a, b) =
1
1
b − a
∫
b
a
dx
x
A(a, b) =
1
b − a
∫
b
a
xd x
G(a, b) = ex p
(
1
Ge − Si
∫
b
a
ln(x)d x
)
M
f
(x
1
, …x
n
) = f
−1
(
1
n
n
∑
i=1
f (x
i
)
)
f (x) = x
p
of 54 90
But what is interesting to me is that to get the geometric mean from the f-mean we have to
change the function f(x) to f(x)=ln(x). This is when it becomes simpler to express the geometric
mean in terms of product notation:"
"
And this is precisely interesting to me because five-fold geometry does a similar thing. We have
a five-fold expression in our AI equation we arrived at:"
"
In that we take the fifth root of the double integral on left. This makes me think of how we can
tile a surface with regular polygons the 3-sided (triangle), 4 sided (square), and 6-sided (regular
hexagon) but five pops out and the pentagon requires another shape added in to tile a surface
without leaving gaps as a so-called Archimedean tessellator, the equilateral triangle, square,
and regular hexagon are the regular tessellators. However, if you are working with solids, there
are five regular solids and they all tile to close off a space, using triangles, for example the
tetrahedron, or squares (the cube), and yes the regular pentagon in the dodecahedron."
See illustration on next page…$
M
0
(x
1
, . . x
n
) =
n
n
∏
i=1
x
i
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
of 55 90
$
of 56 90
It was the Russian scientist Shubnikov who noticed that five-fold symmetry is more
characteristic of life while six-fold symmetry is more characteristic of the physical. He wrote:"
As to the alive organisms, we have not for them theory, which could answer the question what
kinds of symmetry are compatible or incompatible to existence of living material. But we can
note here that remarkable fact that among the representations of the alive nature the
pentagonal symmetry meets more often.
I think from experience and observation you will find this as true if you pay close attention to
Nature. You will find if you look at flowers every now and then you will find six petals around its
center, or sometimes as with a rose perhaps near a hundred petals, but most often you will find
there are five petals around the center of a flower. As well, even in the rose, with near a
hundred petals, they spiral in as a golden spiral, which is built of ratios of the golden ratio (
and use patterns of Fibonacci numbers. The successive ratios between terms in the Fibonacci
sequence converge on at infinity and the golden ratio is derived from pentagonal symmetry
in that if you draw in the chord of a regular pentagon, the ratio of it to its side is . And indeed
the human has two legs, two arms and a head adding up to five, or two eyes, and a nose and a
mouth adding up to five. Or, five fingers, or five toes on each hand or each foot. But for the
physical like a snowflake, there are six points that form around it giving it hexagonal symmetry.
The starfish has five arms."
In looking at life we notice it is based on carbon which is in group 14 of the periodic table of
the elements just like semiconductor elements silicon and germanium. It is because of this that
carbon works because it means has 4 valence electrons, meaning it can form long chains with
hydrogen making organic matter the hydrocarbons, utilizing oxygen (O), nitrogen (N),
phosphorus (P), and sulfur (S). Life does not seem to be based on silicon, though, even though
it has 4 valence electrons as well because while carbon can combine with hydrogen to make
hydrocarbons such as CH4, or combine with O, N, H to make the most simple organic
compound isocyanic acid HNCO which binds H-N=C=O, silicon in the presence of oxygen
forms glass SiO2 so easily that it can not combine with the H, N, C, O, P, and S readily with
each equally so as to form functional hydrocarbons."
It is at this point that I would like to note that carbon is element six in the periodic table giving it
6 protons, and since its molar mass is 12.01, it has 6 neutrons. It so happens that closest
packing of equal radius spheres in the plane like protons, and neutrons is six-around one or
hexagonal symmetry. As Buckminster Fuller constructed his geometry in Synergetics, he
outlined his discovery that equal-radius spheres pack in the form of what he called the vector
equilibrium, which is the cuboctahedron, which he demonstrated was the most transformable
construct and as such becomes pivotal to his Synergetics,"
I would like to suggest in light of this that since carbon has six protons and six electrons, with
the six protons determining its number of electrons (6 to be neutral) giving it four valence
electrons in its outer shell for combining with other elements (the outer shell is four and wants
four to complete an octet, such as four hydrogens each H+, that though life more often meets
with pentagonal symmetry, and here we see carbon meets with six-around-one in the plane, or
twelve-around-one in space as the vector equilibrium, or six-fold symmetry, it is because life is
built out of the physical, like carbon to make the biological, characteristic of pentagonal
symmetry. And it is here I suggest that life animates out of a dynamic structuring of the
physical (inanimate). See illustration on the next page…$
Φ)
Φ
Φ
of 57 90
of 58 90
Indeed we see life could be the interplay between 3, 4, 5, 6 as structured in Buckminster
Fuller’s Synergetics. For instance the vector equilibrium (cuboctahedron) is made of equilateral
triangles and squares, the regular tessellators. With eight triangles and six squares. All of this
speaks respectively of NH3 (ammonia, believed to have contributed to making the amino acids
the building blocks of life) which is three hydrogens around a Nitrogen, CH4 (methane, believed
to have contributed to the formation of amino acids in primordial earth as well) the eight
triangles in the cuboctahedron representing the combination of elements such that they
complete an octet, and its six squares, the six protons, six neutrons, and six electrons of
carbon."
With all said here so far, it might be said that understanding life and its origins can be
understood by looking at artificial intelligence."
Let us return to the geometric mean becoming a different function in the f-mean. We have:"
"
"
p=1 yields:"
"
Is the arithmetic mean between x1 and x2. Now take p=-1:"
="
"
Is the harmonic mean between x1 and x2. Now we try p=0 hoping to get the geometric mean…"
="
="
M
f
= f
−1
(
1
n
n
∑
i=1
f (x
i
)
)
M
f
(x
1
, x
2
) = f
−1
(
1
n
2
∑
i=1
x
p
i
)
p
M
f
(x
1
, x
2
) =
(
1
2
x
1
+
1
2
x
2
)
=
x
1
+ x
2
2
M
f
(x
1
, x
2
) =
(
1
2
2
∑
i=1
x
−1
i
)
−1
2
1
x
1
+
1
x
2
=
2x
1
x
2
x
1
+ x
2
M
f
(x
1
, x
2
) =
(
1
2
2
∑
i=1
x
i
0
)
1
0
(
1
2
2
∑
i=1
1
)
1
0
of 59 90
"
So for we can’t make sense and we have to search for a function that will produce
the geometric mean in the f-mean. It is ln(x). This is interesting because the natural log of x was
created to settle the following conundrum:"
"
This is where we need to create the natural logarithm function so we can have a solution to
such an integral and, we have"
"
Where"
"
"
Let us return to our . It is not a sum"
"
But is a product"
="
"
What this says is that what is important is not the values of data points in an experiment, not
the ’s but the i’s themselves, the number of the data point. Like the one in measurement 1,
the 2 in measurement 2. Never mind that measurement 1 might equal 2.3 grams, measurement
3 might equal 0.5 grams, the important thing is the 1/2 outside the parenthesis because we are
taking, which is is always 1. This is how reality never has any meaning: we
just change to f(x)= ln(x), which is the equivalent of writing"
(
1
2
)
∞
f (x) = x
p
∫
d x
x
=
∫
x
−1
d x =
x
−1+1
0
=
x
0
0
∫
d x
x
= ln(x) + C
ln(x) = log
e
(x)
e = 2.718…
(
1
2
)
∞
1
2
∞
∑
i=1
i
i
=
1
2
(
1
1
+
2
2
+
3
3
+ …
)
M
0
(x
1
, x
2
, x
3
…x
n
) =
1
2
∞
n
∏
i=1
i
i
1
2
(
1
1
⋅
2
2
⋅
3
3
⋅ …
)
x
i
i /i
1/1,2/2,3/3,...
f (x) = x
p
of 60 90
"
Thus it is the experience itself that counts, we find "
"
If we say e=2.718…$
G =
n
n
∏
i=1
x
i
∫
d x
x
= ln(x) + C
of 61 90
!
of 62 90
of 63 90
of 64 90
of 65 90
Miller-Urey
As we can account for some of the amino acids, the building blocks of life, in that Miller and
Urey showed eleven of the twenty amino acids in proteins can be made by mixing together the
primordial earth substances CH4 (methane) NH3 (Ammonia) and H2O water with a few other
gases that were present, and then applying electricity to simulate electrical storms, we need
not just amino acids to have life but sugars for the DNA and RNA that encode life."
Sugars are carbohydrates with the formula:"
"
Where n is 2 to 7. n=1 is not a sugar and is called formaldehyde which is:"
"
Which has the same structure as a sugar, which is a monomer from which the sugars form; that
is, sugars are polymers of formaldehyde. For DNA (deoxyribose nucleic acid) and RNA (ribose
nucleic acid) that encode life we need deoxyribose:"
"
And, ribose:"
"
The sugar produced by plants through photosynthesis that serves for its food is glucose
which is n=6 in . To make these sugars formaldehyde first combines to
make the sugar glyceraldehyde (n=3):"
"
Then combines with this. But to explain the origins of life in terms of arising from a primordial
substance, we need to explain how we make formaldehyde."
Formaldehyde is an intermediate in the combustion or oxidation of methane (one of the
primordial gases that make some of the amino acids). It does not accumulate in the
environment because it is broken down by sunlight or by bacteria in soil and water. It is
produced by the action of sunlight and oxygen on atmospheric methane. In the lab it is stored
as an aqueous solution (formalin) because it polymerizes spontaneously into
paraformaldehyde. It exist in the interstellar medium (the empty space between stars) and is
proposed to be formed there by the hydrogenation of carbon monoxide ice:"
"
"
In order to determine whether life can arise spontaneously or not, a brief review of what we
know suggests what I call an activation function."
C
n
(H
2
O)
n
CH
2
O
C
5
H
10
O
4
C
5
H
10
O
5
C
6
H
12
O
6
C
n
(H
2
O)
n
C
3
H
6
O
3
H + CO ⟶ HCO
HCO + H ⟶ CH
2
O
of 66 90
Miller Urey Chemistry"
Under Nobel prize winner in chemistry (1934) Harold Urey, Stanley Miller a graduate student set
out to see if he mimicked the theoretical primordial Earth, he could produce the 20 amino acids
that are the building block of life. He created the ocean by filling a closed glass container with
water, and coming out of this was a tube that went to a second chamber that mimicked the
primordial earth atmosphere which was methane, ammonia, and hydrogen. As the heated
water vaporized it flowed out of the first chamber and into the second chamber. He passed
electricity through the second chamber to mimic electrical storms or lightning. Between the
first chamber and second he placed a condenser, so when the water vapor rose into it, some of
it would condense into liquid to mimic rain. With this experiment he produced 10 of the 20
biological amino acids. We have yet to find a way to produce all 20 under theoretical primordial
earth conditions."
Produced"
Glycine, Alanine, Aspartate, Valine, Leucine, Glutamate, Isoleucine, Serine, Proline, Threonine"
Not Produced"
Phenyalanine, Tyrosine, Tryptophan, Histidine, Lysine, Arginine, Cysteine, Methionine,
Asparagine, Glutamine"
H2O"
Water is an extraordinary substance and in order to have life you need it. The earth is not only
the right distance from the sun for water to exist in three phases (ice, vapor, and liquid) but is
also happens to be very plentiful here, it covers three quarters of the planet’s surface and what
is more there are great amounts of it under its surface. Let’s look at some of the properties of
water that it has that allows for life:"
1. Water is solvent meaning it dissolves a great number of substances."
2. Water is cohesive and adhesive, cohesive because it flows freely, yet adhesive in that can
also adhere to surfaces. Unicellular organisms rely on external water to transport nutrients
and waste while multicellular organisms have internal vessels that use it to do the same.
Because of adhesion and cohesion water can climb up from the roots of a tree to its top by
tension created by water evaporating from its leaves."
3. Water has a high surface tension meaning plant debris can rest on its surface providing
food and shelter for aquatic life."
4. Water in its solid phase (ice) is less dense than it is in its liquid phase because when it
freezes it expands meaning it floats on the surface water. If it was not for this life could not
exist on earth because if the ice sank the ponds, lakes, and perhaps even the oceans
would freeze over solid."
5. Water has a high heat capacity. The specific heat of water is one calorie per gram degree
centigrade which means it takes one calorie to raise the temperature of a gram of it by one
degree centigrade. This keeps the earth relatively cool, and thus life thrives. A lot of the
sunlight’s energy goes into vaporizing it into clouds that would otherwise go into heating
the planet."
Dehydration Synthesis and Hydrolysis"
Let us look at how water synthesizes substances and breaks them down. Take making fat from
glycerol and a fatty acid:"
of 67 90
If you heat it you remove an H
from the glycerol and an OH
(hydroxide) from the fatty acid
which is to remove a water
molecule H2O leaving an O in the
glycerol and a C in the fatty acid
that joins the glycerol with the
fatty acid to make the fat. For
hydrolysis you add an OH to the
fatty acid and and H to the
glycerol by adding water (H2O) to
the fat thus breaking down it
down into a fatty acid and
glycerol."
DNA and RNA"
Life is encoded by DNA (deoxyribose nucleic acid) and RNA (ribonucleic acid) which make
nucleic acids. Each nucleic acid is a monomer in a polymer called a nucleotide. The monomer
consists of a phosphate, a 5 carbon sugar, and a nitrogen containing base. The phosphate is
the phosphate ion . Deoxyribose is the sugar and is in DNA and ribose is the
sugar . The bases are guanine, cytosine, adenine, thymine in DNA and RNA uses
uracil in place of thymine. The phosphate combined with the sugar is the backbone of DNA
and RNA, and the bases are attached to the backbone. There are two back bones running
parallel to one another and the bases of one attach to the bases of the other in a pairing that is
always guanine (G) pairs with cytosine (C) and adenine (A) pairs with thymine (T). It is the
sequencing of these pairings the encodes for life, and the parallel backbones are twisted so
you have a twisted ladder where the base pairings are its rungs. "
Guanine: "
Cytosine: "
Adenine: "
Thymine: , Uracil: $
PO
4
−
C
5
H
10
O
4
C
5
H
10
O
5
C
5
H
5
N
5
O
C
4
H
5
N
3
O
C
5
H
5
N
5
C
6
H
6
N
2
O
2
C
4
H
4
N
2
O
2
of 68 90
of 69 90
The Problem"
In order to have life we need to have the nucleic acids, which means we need to have the
bases guanine, cytosine, adenine, thymine, and uracil. Can they arise spontaneously from the
calculated conditions of the primordial earth?"
John Oro in 1961 found amino acids and adenine could form from the mixture of hydrogen
cyanide and ammonia in water. Later researchers found several of the bases needed were
present if they allowed hydrogen cyanide to combine with the ammonia when heated in acid.
The problem is hydrogen cyanide present in the laboratory experiment was hundreds of
thousands of times more concentrated than is calculated to have existed on the primordial
earth surface. Further, hydrogen cyanide cannot be concentrated by the evaporation of sea
water in a tidal pool because it is more volatile than water,."
Later, Leslie Orgel found that freezing a hydrogen cyanide solution would allow it to form in the
voids between ice crystals, which meant adenine could only form in the frozen polar regions. In
1975 Miller froze the stuff for 27 years, then analyzed it finding small amounts of several of the
bases including the adenine."
We now know that four molecules of hydrogen cyanide can combine to form
diaminomaleonitrile, then, under sunlight, if it reacts with another molecule of hydrogen
cyanide it produces adenine in 7% yield. But if four molecules react with salt ammonium
formate there is 90% yield of adenine. However, this requires dehydration, by removing two
molecules of water, meaning we need to boil away the water of the solution to dryness."
The Solution"
Substances and rapid temperature changes (cold for some reactions, warm for others like
dehydration synthesis) that are not present today, nor that we calculate were present in the
primordial earth (like sufficient quantities of hydrogen cyanide) that are needed to account for
a prebiotic pathway to the nucleobases and their combination with phosphates and sugars,
and all of the biological amino acids would suggest there was the presence of what I will call an
activation function. Since these necessary substances are not present today, and cannot be
calculated to have existed a long time ago, I suggest that the activation function was a limiting
factor, that as it activated life from what was present, it depleted determining how much life
was present in the beginning by its total depletion. Once life exists the production of the
nucleic acids is possible because it can now be powered by life’s consumption of
carbohydrates, which only can exist on the earth after the existence of life."
The Activation Function"
I would like to suggest that the prebiotic chemistry might have been passed through an
activation function that disappeared after life was on its way to evolving. "
The problem, then, of answering the question of how life began is one of finding the activation
function and its mechanism by which it takes prebiotic chemistry and activates it (makes it
alive) so it can now self-replicate, and evolve. We assume that as this mechanism activates the
molecules, its mechanism depletes as it activates from what is available. In this sense the
mechanism is a limiting reactant, so it determines how much material is activated before it
depletes completely."
of 70 90
Logically, the way to determine what this mechanism is, and how it serves as an activation
function is to look for the by-products of the reaction that are left over, and from that, deduce
its nature."
To do this, we have to look for that thing in our knowledge of the Earth’s history that does not
make sense. This would be in the faint young star paradox. We know that five billion years ago,
when the Earth and Sun first formed, that the sun was 0.7 times its present output and so, the
Earth should have been frozen over, yet, we know it was not. That it had water in its liquid
phase. Thus something was there that is not present today. That something must have been
the mechanism for the activation function that “turned on” prebiotic chemistry."
If is the activation function, where x is the prebiotic material, and we say is residue of
the reaction, and is the activated substance (life) then,"
"
We know . If we can find in nature, we can deduce ."
I have presented it like this because 1) Life has not been created in the laboratory from scratch
2) New life does not seem to be originating on earth in present times. Therefore, the activation
function is probably not present on Earth today and more than likely disappeared, or depleted
after activating prebiotic chemistry. Life exists, yet we do not know how prebiotic substances
organize into self-replicating systems that evolve. Therefore, we must look for something
concerning the Earth that does not make sense. I suggest that would be the young star
paradox. If the Earth had water in its liquid phase when it should have been frozen over, then
something could have existed then that was a limiting reactant, or something like it, that
activated prebiotic substances, in that it was responsible for warming the earth (perhaps a heat
retaining substance)."
I use the term limiting reactant loosely as well as prebiotic chemistry because one, the reactant
was not necessarily a substance alone, but a manifestation of energy not just necessarily
sunlight incident upon the earth and, the prebiotic chemistry was not necessarily just
substances that existed then from what we have theoretically calculated.$
σ (x)
r
l
l + r = σ (x)
l
r
σ (x)
of 71 90
Bone As A Mathematical Construct
What better place to begin than with than bone as it is the basic framework around which skeletal life is
structured, the vertebrates. Here is what I found in bone as a mathematical construct:
In my exploration of the connection between biological life and AI the most dynamic component is that of
bone. It affords us the opportunity to look at:
Multiplying Binomials
Completing The Square
The Quadratic Formula
Ratios
Proportions
The Golden Ratio
The Square Root of Two
The Harmonic Mean
of 72 90
Density of silicon is Si=2.33 grams per cubic centimeter.
Density of germanium is Ge=5.323 grams per cubic centimeter.
Density of hydroxyapatite is HA=3.00 grams per cubic centimeter.
This is
where
Where HA is the mineral component of bone, Si is an AI semiconductor material and Ge is an AI
semiconductor material. This means
The harmonic mean between Si and Ge is HA,…
This is the sextic,…
Which has a solution
Where x=Si, and y=Ge. It works for density and molar mass. It can be solved with the online Wolfram
Alpha computational engine. But,…
3
4
Si +
1
4
G e ≈ H A
H A = Ca
5
(PO
4
)
3
OH
Si
H A
Si +
[
1 −
Si
H A
]
G e = H A
2 SiG e
Si + G e
≈ H A
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
G e
=
1
2 + 1
1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
Si =
1
2
G e
±
H A
G e
H A
2
−
4G e
H A
+ 4
Si = G e − H A
of 73 90
Si
H A
Si +
[
1 −
Si
H A
]
G e = H A
Si
2
H A
+ Ge −
Si
H A
G e ≈ H A
1
H A
Si
2
−
G e
H A
Si + G e ≈ H A
1
H A
2
Si
2
−
G e
H A
2
Si +
G e
H A
≈ 1
1
H A
2
Si
2
−
G e
H A
2
Si +
G e
H A
− 1 ≈ 0
1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
of 74 90
We see that the square of the binomial is a quadratic where the third term is the square of one half the
middle coefficient. This gives us a method to solve quadratics called completing the square:
(x + a)(x + a) = x
2
+ 2a x + a
2
(x + a)
2
= x
2
+ 2a x + a
2
a x
2
+ bx + c = 0
a x
2
+ bx = − c
x
2
+
b
a
x = −
c
a
(
1
2
b
a
)
2
=
1
4
b
2
a
2
x
2
+
b
a
x +
1
4
b
2
a
2
= −
c
a
+
1
4
b
2
a
2
(
x +
1
2
b
a
)
2
=
b
2
− 4a c
4a
2
x +
b
2a
=
±
b
2
− 4a c
2a
x =
−b
±
b
2
− 4a c
2a
of 75 90
1
H A
2
Si
2
−
G e
H A
2
Si +
[
G e
H A
− 1
]
= 0
x =
−b
±
b
2
− 4a c
2a
a =
a
H A
2
b = −
G e
H A
2
c =
[
G e
H A
− 1
]
b
2
− 4a c =
G e
2
H A
4
− 4
1
H A
2
[
G e
H A
− 1
]
=
G e
2
H A
4
−
4G e
H A
3
+
4
H A
2
=
1
H A
2
[
G e
2
H A
2
−
4G e
H A
+ 4
]
b
2
− 4a c =
1
H A
(
G e
H A
− 2
)
2
x =
Ge
HA
2
±
1
HA
[
Ge
HA
− 2
]
2
HA
2
=
1
2
G e
±
1
2
H A
[
G e
H A
− 2
]
=
1
2
G e
±
1
2
G e − H A
Si =
1
2
G e +
1
2
G e − H A
Si = G e − H A
of 76 90
A ratio is and a proportion is which means a is to b as b is to c.
The Golden Ratio
and.
or
Si ≈ G e − H A
H A ≈
2 SiG e
Si + G e
Si ≈ G e −
2 SiG e
Si + G e
(Si + G e)G e
Si + G e
−
(Si + G e)Si
Si + G e
−
2 SiG e
Si + G e
= 0
G e
2
− 2SiG e − Si
2
Si + G e
= 0
x
2
− 2x y − y
2
= 0
x
2
− 2x y = y
2
x
2
− 2x y + y
2
= 2y
2
(x − y)
2
= 2y
2
x − y =
±
2y
x = y + 2y
x = y(1 + 2)
x
y
= 1 + 2
y
x
=
1
2 + 1
Si
G e
≈
1
2 + 1
a
b
a
b
=
b
c
(
Φ
)
a
b
=
b
c
a = b + c
a c = b
2
c =
b
2
a
of 77 90
The mineral component of bone hydroxyapatite (HA) is
The organic component of bone is collagen which is
We have
a = b +
b
2
a
b
2
a
− a + b = 0
b
2
a
2
− 1 +
b
a
= 0
(
b
a
)
2
+
b
a
− 1 = 0
(
b
a
)
2
+
b
a
+
1
4
= 1 +
1
4
(
b
a
+
1
2
)
2
=
5
4
b
a
= −
1
2
±
5
2
b
a
=
5 − 1
2
a
b
=
5 + 1
2
ϕ =
5 − 1
2
Φ =
5 + 1
2
ϕ =
1
Φ
Ca
5
(PO
4
)
3
OH = 502.32
g
m ol
C
57
H
91
N
19
O
16
= 1298.67
g
m ol
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
= 0.386795722
ϕ = 0.618033989
1 − ϕ = 0.381966011
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
of 78 90
%
We said bone was characterized by the sextic:
Which has a solution
Now let us look at our Si wave equation. We said
And we have
0.381966011
0.386795722
100 = 98.75
Si
G e
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
G e
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
G e
=
1
2 + 1
y
x
=
1
2 + 1
∇ ×
B = μ
J + μϵ
0
∂
E
∂t
Si
ΔGe
ΔS
= B − Ge
ΔSi
ΔS
ΔGe =
ΔS
Si
B −
Ge
Si
ΔSi
(
∇
2
−
1
c
2
∂
2
∂t
)
E = 0
(
∇
2
−
1
c
2
∂
2
∂t
)
B = 0
c =
1
ϵ
0
μ
≈ ϕ
of 79 90
We see and are both and c is in the Si (silicon) field wave, but for E and B fields c is the speed of
light.
Connecting The Two
We can write
This is the quadratic
Which has solutions
We can say…
μ
ϵ
0
Φ
ϕ
ϵ
0
= 8.854E − 12F ⋅ m
−1
μ = 1.256E − 6H /m
Ge
Si
= μϵ
0
ΔS
Si
= μ
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Si = 0
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Ge = 0
G e − Si
Si
= Φ
1
Ge
Si
= ϕ
x
2
− 3x y + y
2
= 0
y =
(
3
2
−
5
2
)
x = 0.381966x
y =
(
3
2
+
5
2
)
x = 2.618033989x
G e = (ϕ + 1)Si
of 80 90
From the case we made from the Si wave equation:
And we can say
From the case we made for bone:
Thus we have two approximations for Si/Ge. Just how far apart are they? This is x:
Is in the ratio between successive integers 1, 2, ,3, 4, 5.
We said bone was characterized by the sextic:
Which has a solution
Now let us look at our Si wave equation. We said
And we have
x
2
− 3x y + y
2
= 0
G e = ( 2 + 1)Si
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
2x =
5 + 1
2
x =
143
125
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
G e
=
1
2 + 1
y
x
=
1
2 + 1
∇ ×
B = μ
J + μϵ
0
∂
E
∂t
Si
ΔGe
ΔS
= B − Ge
ΔSi
ΔS
ΔGe =
ΔS
Si
B −
Ge
Si
ΔSi
of 81 90
We see and are both and c is in the Si (silicon) field wave, but for E and B fields c is the speed of
light.
Connecting The Two
We can write
This is the quadratic
Which has solutions
(
∇
2
−
1
c
2
∂
2
∂t
)
E = 0
(
∇
2
−
1
c
2
∂
2
∂t
)
B = 0
c =
1
ϵ
0
μ
≈ ϕ
μ
ϵ
0
Φ
ϕ
ϵ
0
= 8.854E − 12F ⋅ m
−1
μ = 1.256E − 6H /m
Ge
Si
= μϵ
0
ΔS
Si
= μ
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Si = 0
(
∇
2
−
1
ϕ
2
∂
2
∂x
)
Ge = 0
G e − Si
Si
= Φ
1
Ge
Si
= ϕ
x
2
− 3x y + y
2
= 0
of 82 90
We can say…
From the case we made from the Si wave equation:
And we can say
From the case we made for bone:
Thus we have two approximations for Si/Ge. Just how far apart are they? This is x:
Is in the ratio between successive integers 1, 2, ,3, 4, 5.
y =
(
3
2
−
5
2
)
x = 0.381966x
y =
(
3
2
+
5
2
)
x = 2.618033989x
G e = (ϕ + 1)Si
x
2
− 3x y + y
2
= 0
G e = ( 2 + 1)Si
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
2x =
5 + 1
2
x =
143
125
of 83 90
The Idea of Mathematical Construct
In my works The Mathematical Nature of Life (Beardsley 2021) and Perfect Equations Beardsley (2021) I
set out to find if the the elements and compounds characteristic of life and artificial intelligence (AI) do
not just conform to chemical law, but if they are purely mathematical independently of the use of
chemistry to describe them, and if they are connected to one another. The simplest example of this for
biological life and AI would be that the most basic organic compound is HNCO (isocyanic acid) where H
(hydrogen), N (nitrogen), C (carbon), and O (oxygen) are the most abundant biological elements. Indeed
biological elements are for the most part organic, which means they are made of long chains using carbon
with hydrogen, which they can form because C is C4- and H is H+ meaning we can have:
And in isocyanic acid we have:
H-N=C=O
Where H is H+, N is N3-. C is C4-, O is O2-, the H uses its single bond with one from nitrogen, leaving
N2- or two bonds which go to C leaving for it C2- which goes to oxygen that needs it because it is O2-.
Thus all is satisfied by chemical law. In my search for mathematical law, I find it exists in the case of
HNCO and the AI semiconducting element silicon (Si) and its doping agents P and B as such (by molar
mass):
This paper strives to break down such mathematical equations for biological life and artificial intelligence
into their components to find what is acting to create such constructs. In the second book I actually
brought the planets into the mix with some very interesting results. As another example, water and air, the
main physical constituents that interact with life we have:
C + N + O + H
P + B + Si
≈ ϕ
ϕ =
a
b
=
5 − 1
2
c = b + a
a
b
=
b
c
H
2
O
air
≈ ϕ
air = 0.25O
2
+ 0.75N
2
of 84 90
By molar mass for air as a mixture (not a compound). With this air is 29.0 grams per mole.
Molecular Geometry
We will want to break down our equations into the components of their geometric relationships and see if
they predict the bond angles of some of the basic substances considered. We will look here at linear,
trigonal planar, and tetrahedral.
Linear, like CO2 (carbon dioxide) its bond angle is 180 degrees:
Trigonal planar, like SO3 (sulfur trioxide) its bond angle is 120 degrees:
That is, S is at the center and the O atoms are 120 degrees apart due to the even division of 360/120=3.
Tetrahedral, like methane (CH4) one of the the primordial gases that may have contributed to making
some of the amino acids, the building blocks of life as show by Miller and Urey in the early origins of
life:
This is 109.5 degrees apart from arcos (1/3) = 109.5
But what if we are considering not just neutral molecules but polyatomic anions that have a net charge. In
such instances, the free electron pairs compress the expected 120 degree bond angle in the atoms around
the central atom to 115 degrees as with the nitrite ion NO2-:
of 85 90
Similarily we have for O3 (ozone) that the bond angle is 116 degrees in its deviation from 120 degrees.
The configuration is:
Both of these anions are important to the life and the theory of how life forms. O-zone is more of a
physical component in that in the stratosphere it absorbs UV radiation harmful to life.
Breaking Down Bone
Essentially, in our mathematical formulation of bone, we had that
Which resulted in that the AI elements:
By way of the mineral component of bone HA (hydroxyapatite) is the harmonic mean between Silicon
and Germanium the primary semiconductor elements, which are really the skeleton on AI. Thus, we need
to break down the harmonic mean between Si and Ge into its geometric representation, and through find
what its components are if we are to get any sense of the dynamics. Here I do that in the following
illustration…
Si ≈ G e − H A
H A ≈
2 SiG e
Si + G e
Si
G e
≈
1
2 + 1
of 86 90
of 87 90
We see that through bone Si and Ge predict an angle of about 116 degrees. This is not the case of linear at
180 degrees, or tetrahedral pyramidal at 109.5 degrees, but is the instance of trigonal planar, but not of
neutral molecules, which is 120 degrees, but of trigonal planar for polyatomic anions such as the nitrite
ion:
And O-zone (Not an anion but has free electrons due to a single bond):
of 88 90
The Masculine and Feminine
Here I will suggest the term masculine silicon and feminine germanium in place of positive (p-type
silicon) and negative (n-type germanium) respectively. And, I will denote them , and , which are dagger
and double dagger.
We say since silicon (Si) doped with boron (B) is p-type silicon because boron being in group 13 only has
three valence electrons and silicon wants four, giving it a deficiency of negative electrons and thus a net
positive distribution that can carry electrons, holes they can fall into. Thus I will say:
And since we say germanium (Ge) doped with phosphorus (P) is n-type silicon because phosphorus being
in group 15 has five valence electrons and germanium, being in group 14 like silicon, wants four
electrons. Thus it has a surplus of negative electrons and thus a net negative distribution that can carry a
current. Thus I will say:
Since B/Si=10.81/28.09=0.3867 and Ge/P=72.64/30.97=2.345 we have:
and.
Now we turn from this construct of the masculine and feminine in AI to the masculine and feminine in
biology.
We consider the female sex hormone estradiol (estrogen , E):
And the male sex hormone testosterone (T):
And, cholesterol (Ch) from which both are made:
And notice,…
And we consider the semiconductor materials used to make AI:
†
‡
B
Si
= †
G e
P
= ‡
† = 0.3867
‡ = 2.345
C
18
H
24
O
2
= 272.38g /m ol
C
19
H
28
O
2
= 288.42g /m ol
C
27
H
46
O = 386.65g /m ol
Ch + T
E
= 2.5
G e
Si
= 2.6
of 89 90
And write,…
We notice that the masculine (T) is in inverse relation to the feminine (E), but that the two add up to on
whole (Ch) in that the masculine has coefficient 1-Si/Ge and the feminine has coefficient 1-Ge/Si. This
expresses the inverse relationships between man and woman.
I interpret this as the masculine (T) is in inverse relation to the feminine (E), but that the two add up to a
whole (Ch) in that the masculine has coefficient 1-Si/Ge and the feminine has coefficient 1-Ge/Si that is
they are inverse relation but compliment one another. How would an AI use this information to determine
its sex?…
The male is reduced less in the difference between 1 and Si/Ge, but the the female is reduced less by
having Ge in the numerator. It is really quite egalitarian.
We now see that:
And this shows the connection of masculine and feminine AI to masculine and feminine biological life.
Ch + T
E
=
G e
Si
T =
G e
Si
E − Ch
E =
Si
G e
(T + Ch)
T
(
1 −
Si
G e
)
+ E
(
1 −
G e
Si
)
= Ch
(
Si
G e
− 1
)
T
(
1 − †
)
+ E
(
1 − ‡
)
= Ch
(
Si
G e
− 1
)
of 90 90
The Author